Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$
Problem
Source: CWMI 2018 Q7
Tags: number theory, least common multiple, inequalities
MarkBcc168
17.08.2018 05:39
Take $n=1$ and $m=p^c$ .
J_bucca
17.08.2018 07:36
MarkBcc168 wrote: Take $n=1$ and $m=p^c$ . In this case $\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p$ doesn't satisfy the condition.
primes020
17.08.2018 15:20
take $m=p^{2c-1}$, $n=p^{2c-1}-p^{c-1}$
zsgvivo
22.07.2019 10:05
take $m=p^{c+1}$and$n=m-p$
laikhanhhoang_3011
29.05.2021 13:03
zsgvivo wrote: take $m=p^{c+1}$and$n=m-p$ Can you explain why you choose those number? I cant understand
ilovemath0402
07.09.2024 03:49
zsgvivo wrote: take $m=p^{c+1}$and$n=m-p$ Why this satisfied the requirement, i see that the ratio is less then $p$